Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}$ is equal to :

• A. $\log a$
• B. $\log 2$
• C. $aa$
• D. $\log x$

Answer 1

Answer: (A)

$\log a$

$\underset{x\to \frac{\pi }{2}}{lim}$ $\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}$

$=\underset{x\to \frac{\pi }{2}}{lim}$ $a^{\cos x}$ $\frac{[\frac{a^{\cot x}}{a^{\cos x}}-1]}{\cot x-\cos x}$

$=\underset{x\to \frac{\pi }{2}}{lim}$ $a^{\cos x}$ $\left[\frac{a^{\cot x-\cos x}-1}{\cot x-\cos x}\right]$

$=\log a$ ........ $(\because \underset{x\to 0}{lim}\frac{a^x-1}{x}=\log a)$